Questions tagged [copula]
A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.
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Calculation on Quadrant Probability for Bivariate data using Copula
I have a question regarding the computation on bivariate probability when using copula function.
Let $u=F_X(x)$ and $v=F_T(t)$ be the CDF for marginals of $X$ and $T$ respectively, a joint ...
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Implied Correlation from Gaussian Copula
I am building a spreadsheet model that allows marginal distributions to be correlated together using a Gaussian copula (with prescribed correlation matrix).
The inputs into the model are the ...
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Indefinite integral | How to compute the second-order partial derivatives of Mixed (2 gaussian & 2 binary) Gaussian Copula?
This is a problem about "Computing the Mixture Gaussian Copula with 2 normal (continuous) variables and 2 binary (discrete) variables".
Problem background
I have two continuous r.v. and two ...
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How is the copula $C\big(X_1, 1/X_1, \exp(-X_1)\big)$ equal to the copula $C(X_1, - X_1, -X_1)$ and what is its form?
Let $X_1$ be a positive random variable with a continuous cumulative distribution function and $C$ the copula of $(X_1, 1/X_1, \exp(-X_1))$.
Why is $C$ also the copula of $(X_1, -X_1, -X_1)$ and what ...
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Generate uniform random numbers with a certain dependency structure
Let's assume I wanted to sample from a $\mathcal N(\boldsymbol 0, \boldsymbol\Sigma)$ distribution, where $$\boldsymbol\Sigma = \begin{pmatrix} 1 & \rho \\ \rho & 1\end{pmatrix}$$ for some $\...
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Stochastic ordering of dependent random variables given their sum and a common copula
I am interested in pointers for the stochastic ordering of continuous dependent variables with common copulas.
I have a vector $X=(X_1,X_2,...,X_N)$ and a vector $Y=(Y_1, Y_2, ..., Y_N)$. X and Y ...
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Does uniform distribution on every square $[0,a]^2$ along diagonal imply uniform CDF on the entire $ [0,1]^2$
Let $F: [0,1]^2\to R $ be a continuous cdf with uniform marginals, i.e., $F(x,1)=x$ and $F(1,y)=y$. Suppose $F$ is symmetric, i.e., $F(x,y)=F(y,x)$.
Suppose we also know that $F(a,a)=a^2$ for all $a\...
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How to prove that the multivariate Frechet-Hoeffding lower bound is not copula when $n \geq 3$?
I want to show that the multivariate Frechet-Hoeffding lower bound given by $C^-(u_1,\dots, u_n) = \left(\sum_{i=1}^{n}{u_i} - (n-1)\right)_+$ is not a copula when $n \geq 3$.
My question is:
Is it ...
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Semi-survival copula
I'm currently doing some reading on copula function. For a bivariate cdf, by Sklar's theorem, there exists a copula function,C, such that
$P(U<u,V<v) = C(u, v)$.
As written by Nelsen (2006), A ...
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show a function defines a copula
so i got the following problem:
consider the follow.
$$\overline{C}_{\overline{X}}:\mathbb{R}\to [0,1], u\mapsto\overline{C}_{\overline{X}}(u) = \mathbb{P}(\overline{F}_{X_1}(X_1)\leq u_1,...,\...
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Multivariate t distribution: Find probability of region enclosed by constant-density hypersurface
I am working with a multivariate t distribution, say of dimension p. Given a point P = (x1, ..., xp) in the sample space I need to calculate the probability of the region of the sample space enclosed ...
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Monotonicity of Parametric Bivariate Copula w.r.t. $\theta$
Let $C(u_1,u_2;\theta)$ be a bivariate parametric copula.
I know if $C(u_1,u_2;\theta)$ is the Gaussian copula, then $\partial_{\theta} C(u_1,u_2;\theta)>0$ for any $u_1$ and $u_2$ (it follows from ...
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How do I derive a pair-copula decomposition for a joint density function?
In Section 4.1 of Analyzing Dependent Data with Vine Copulas (Czado), the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’...
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Why can this joint distribution function be written as this integral of a conditional distribution function?
In Section 3.8 of Dependence Modeling with Copulas (Joe), the author starts with the following.
In this section, we show how Sklar's theorem applies to a set of univariate conditional distributions, ...
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Finding a value $\alpha$ such that a function $c$ is a copula density of $(U_1, U_2)$
Let
$$c(u_1,u_2) = 1 + \alpha(1- 2u_1)(2- 2u_2)$$
where $u_1, u_2 \in (0,1)$.
The question is, for which $\alpha$ is the function $c$ a copula density for $(U_1, U_2)$.
So my idea was to compute the ...
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Beyond vine copulas
I know that vine copulas allow for different dependency models between pairs of variables. I wondered if they are also used to model dependency models between subsets (beyond pairs) of variables or if ...
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Sum of dependent random variables and copulas
I have two dependent continuous random variables (RVs) $X$ and $Y$ and I'm interested in determining the CDF of the sum, i.e., $F_{X+Y}(t) = \mathbb{P}(X+Y \leq t)$. I know the marginal of $X$ and $Y$ ...
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Proving that copulas are Lipschitz continuous
A copula is a function $C:[0,1]^2\to[0,1]$ such that $C(x,0)=C(0,x)=0$ for all $x\in[0,1]$, $C(x,1)=C(1,x)=x$ for all $x\in[0,1]$, and
\begin{equation}\label{ineq}
C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(...
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Calculating the joint cumulative distribution function from a junction tree
Assume I have the following Junction Tree between random variables $X_1,\dots,X_7$ that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ...
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Product of correlated random variables and its transformation
There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given ...
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Combining conditional probabilities using an unconditional copula
First: just a bit of background on copulas.
Suppose we have a pair of continuous random variables $Y_1, Y_2$ with distribution functions $F_1(y_1)=P(Y_1\leq y_1)$ and $F_2(y_2)=P(Y_2\leq y_2)$.
Let ...
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Proving that $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$ is a copula
I have the following problem. I need to prove the copula property
for the function $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary.
A copula is defined as a ...
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Frequently used methods to estimate copula of discrete random variable in high dimension
I am new to the estimation of copula with discrete marginal distributions. For example, I have the data of 30 discrete random count variables from $X_1$ to $X_{30}$, some $X_j$ has Poisson ...
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Kendall’s tau between $X$ and $1/Y$
I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples.
I'm struggling with this exercise from chapter 8:
...
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Proof that Pearson correlation is/isn't a functional of the copula for a given pair of random variables
I have a growing interest and respect for the subject of copulas initially thanks to comments made on stats.SE by kjetil-b-halvorsen. The most interesting to me right now is the following:
"[T]...
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Sampling from Gaussian Copula with a conditional distribution approach
In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function $C$ being the joint distribution. This procedure ...
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Showing that the lower bivariate Fréchet-Hoeffding bound is a copula.
I want to show that the bivariate Fréchet-Hoefdding lower bound is indeed a copula.
The bivariate function is defined by: $W(x,y)=\max\{x+y-1,0\}, \ x,y\in[0,1] $
Definition "Copula":
A two-...
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How to deal with copulas?
I want to model a couple of variables $(X,Y)$ using copulas. My idea is to model the marginal distributions and then use copulas to combine marginal distributions and copula to get a joint ...
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Kendall's tau for archimedian copula
Consider a 2-dimensional archimedian copula with generator $\psi$ and $\gamma:=\psi^{-1}$, i.e.
$$C(u,v)=\gamma(\psi(u)+\psi(v))$$
I want to show that $$\tau=1-\int_{0}^{\infty}t\gamma'(t)^2dt$$
I ...
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Find $\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)$ where $X,Y$ poisson
Let $X=Y_1+N,Y=Y_2+N$ where $N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2)$ and $N,Y_1,Y_2$ are independent. I'm asked to find $$\lim_{x\uparrow 1}\mathbb{...
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$P(u_1\leq\alpha,u_2\leq\beta)$ and $P(u_1>\alpha,u_2>\beta)$ and copulas
I know this is meant to be elementary, but I don't know why I cannot convince myself -- given a copula $C(u_1,u_2)=P(u_1\leq \alpha,u_2\leq\beta)$, why is $P(u_1>\alpha,u_2>\beta)=1-\alpha-\beta-...
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Verifying if a function is 2-increasing
In the book An Introduction to Copulas (2006), by Roger B. Nelsen, on page 8, the author defines a function $H$ as 2-increasing if
\begin{equation} V_H(B)=H(x_2,y_2)-H(x_2,y_1)- H(x_1,y_2)+H(x_1,y_1)\...
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How can I get an intuition about Copula?
I am really struggling with getting an intuition about copulas. I have red many articles and I am stuck at what is the concept/idea behind it. For example if I have two random variables X and Y and I ...
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Correlation matrix of Gaussian copula
The question is related to the Gaussian copula.
Let $\Phi(x)$ denote the cdf of standard Normal distribution. Let $(X_1, X_2) \sim \mathcal{N}(0,\Sigma)$ be joint Normal with covariance matrix
$$
\...
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Joint Distribution of Two Dependent Variables having the Marginals
From two Independent Normal Random Variables: $X \sim N(\mu_1,\sigma) $ and $Y\sim N(\mu_1,\sigma)$, I created two DEPENDENT random variables $Z$ and $W$:
$Z= X - Y$
$W= X - g(Y)$ where $g(\...
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How to compare the entropy of copulas?
Three different bivariate copulas are shown below with increasing degrees of dependence (parameter $\theta$).
Differential entropy is a measure of disorder in a probability density like the copula. ...
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Impact of copula choice on quantiles (sum of random variables)
I am trying to get my head around the impact of different dependence structures (copulas) on the risk (quantiles) of a sum of dependent random variables (with arbitrary marginals).
In a multivariate ...
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Why is copula entropy negative?
Most statistical measures are non-negative. Shannon entropy, $H(X)$, a statistical measure of disorder or uncertainty in probability distributions, is also non-negative, despite it having a negative ...
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Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?
$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$
is the ...
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How to simplify out of the gamma function $\Gamma(z)$?
$$\mathrm{d}t(x_i; \nu) = \frac{\Gamma \bigg(\frac{\nu+1}{2}\bigg) }{\Gamma (\frac{\nu}{2}) \sqrt{\pi\nu}} \Bigg( 1+\frac{x_i^2}{\nu} \Bigg)^{-\frac{\nu+1}{2} } \enspace, i=1,2$$
How to derive a ...
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Derivation of bivariate Gaussian copula density
The multivariate Gaussian copula density, derived here, is
$$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$
where $\Sigma$ is the covariance ...
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Integrate $\int_{0}^{1} u^{\frac{\alpha}{\beta}-\alpha}(1-\alpha) \log \left((1-\alpha) u^{-\alpha}\right) d u $
Where $c(u, v)=u^{-\alpha}(1-\alpha) $ is the Marshall-Olkin copula density, we have the following integral:
\begin{align}
I_{1} &=\iint_{E} c(u, v) \log c(u, v) d u d v \\
&=\int_{0}^{1} \...
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Entropy of the bivariate Gaussian copula: Closed-form analytical solution
Background
The bivariate Gaussian copula function is
$$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy.$$
The multivariate Gaussian ...
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How to simplify function to $\log (1-\theta)+\frac{\theta}{2-\theta}$?
\begin{align}
f(\theta)& = \log \frac{1}{1-\theta}-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}} \\
& = \log (1-\theta)+\frac{\theta}{2-\theta}
\end{align}
How are the two ...
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Do copulas preserve uniform convergence?
Suppose I have a distribution $Q$ of $\left(Y_i, X_i\right)$, such that $Q(y,x)=C(F_Y(y), F_X(x))$, for some copula $C$ and marginal CDFs $F_X$ and $F_Y$, by Sklar's theorem.
Now, suppose I have ...
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Showing that a function is a (family of) copula(s)
I am currently working with the book An Introduction to Copulas by Roger Nelsen. I am having trouble solving one of the exercises in the first chapter. The exercise reads as follows:
Let $C$ be a ...
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Simplify $\mathbb{E}\left[ -\log c(u,v)\right] $, the expected logarithm of the copula density
2020 11.26: I finished the derivation but haven't posted it here. I leave this open so that those willing to try their own version can if they want
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How can we derive a closed-form analytical ...
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Archimedean Clayton copula entropy
Question
I would like to derive the entropy of Archimedean parametric copulas (Clayton, Frank, or Gumbel), focusing here on the Clayton copula.
Link to similar question on the t-copula.
The bivariate ...
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Gluing copulas along a curve
Let $C(u,v)$ be a 2-copula and $W(u,v)=\max(u+v-1,0)$. I want to glue $C$ with $W$ along the curve $u^2+v^2=1$ s.t $$D(u,v)=\begin{cases}u+v-1,& u^2+v^2\ge 1,\\ C(u,v), &u^2+v^2<1 \end{...
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Find a joint distribution function with known Kendall's tau (Copula)
Assume we have two random variables $X_1\sim Exp(2)$, $X_2\sim Exp(1/2)$. I need to find a joint distribution function, so that Kendall's Tau will be $\rho_\tau(X_1,X_2)=-0.85$. I know that somehow I ...
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